Random Triangles III

Similar Triangles, Another Trace of the Golden Ratio

In this paper we investigate similar triangles which are not congruent but have two pair congruent sides. We show that greatest lower bound of values for similarity ratio of such triangles is golden ratio. For right triangles, we prove that the supremum of values for similarity ratio is the square root of the golden ratio.

A zero-one law for the existence of triangles in random key graphs

Random key graphs are random graphs induced by the random key predistribution scheme of Eschenauer and Gligor under the assumption of full visibility. For this class of random graphs we show the existence of a zero-one law for the appearance of triangles, and identify the corresponding critical scaling. This is done by applying the method of first and second moments to the number of triangles i...

A local central limit theorem for triangles in a random graph

In this paper, we prove a local limit theorem for the distribution of the number of triangles in the Erdos-Renyi random graph G(n, p), where p ∈ (0, 1) is a fixed constant. Our proof is based on bounding the characteristic function ψ(t) of the number of triangles, and uses several different conditioning arguments for handling different ranges of t.

The Ill-Balanced Triangles of the Islamic States in Crisis

Divide and conquer martingales and the number of triangles in a random graph

The goal of this paper is to present a novel application of a recent and useful martingale inequality. As an illustration, we prove an essentially sharp bound for the probability that a random graph contains significantly more triangles than expected.